Simplify the following complex expressions. See Examples 2, 3, and 4. (4-2i)-(3+i) 10. (4-i)(2+i) 11. (3-i)^2 12. i^7 13. (7i-2)+(3i^2-i) 14. (3+i)(3-i) 15. (5-3i)^2 16. (5+i)(2-9i) 17. i^13 18. (9-4i)(9+4i) 19. 11i^314 20. i^132 21. (7-3i)^2 22. (4-3i)(7+i) 23. (3i)^2

1. Simplify $(4-2i)-(3+i)$<br /> Subtract real and imaginary parts separately: $(4-3) + (-2i-i) = 1 - 3i$.<br /><br />2. Simplify $(4-i)(2+i)$<br /> Use distributive property: $4(2+i) - i(2+i) = 8 + 4i - 2i - i^2$. Since $i^2 = -1$, it becomes $8 + 2i + 1 = 9 + 2i$.<br /><br />3. Simplify $(3-i)^2$<br /> Apply the formula $(a-b)^2 = a^2 - 2ab + b^2$: $(3-i)^2 = 3^2 - 2(3)(i) + (i)^2 = 9 - 6i - 1 = 8 - 6i$.<br /><br />4. Simplify $i^7$<br /> Recognize powers of $i$: $i^7 = i^{4+3} = (i^4)(i^3) = 1 \cdot (-i) = -i$.<br /><br />5. Simplify $(7i-2)+(3i^2-i)$<br /> Combine like terms and use $i^2 = -1$: $(7i - 2) + (3(-1) - i) = 7i - 2 - 3 - i = -5 + 6i$.<br /><br />6. Simplify $(3+i)(3-i)$<br /> Use difference of squares: $(3+i)(3-i) = 3^2 - (i)^2 = 9 - (-1) = 10$.<br /><br />7. Simplify $(5-3i)^2$<br /> Apply the formula $(a-b)^2 = a^2 - 2ab + b^2$: $(5-3i)^2 = 5^2 - 2(5)(3i) + (3i)^2 = 25 - 30i - 9(-1) = 34 - 30i$.<br /><br />8. Simplify $(5+i)(2-9i)$<br /> Use distributive property: $5(2-9i) + i(2-9i) = 10 - 45i + 2i - 9i^2$. Since $i^2 = -1$, it becomes $10 - 43i + 9 = 19 - 43i$.<br /><br />9. Simplify $i^{13}$<br /> Recognize powers of $i$: $i^{13} = i^{12+1} = (i^4)^3(i) = 1 \cdot i = i$.<br /><br />10. Simplify $(9-4i)(9+4i)$<br /> Use difference of squares: $(9-4i)(9+4i) = 9^2 - (4i)^2 = 81 - 16(-1) = 97$.<br /><br />11. Simplify $11i^{314}$<br /> Recognize powers of $i$: $i^{314} = (i^4)^{78}i^2 = 1 \cdot (-1) = -1$. Thus, $11i^{314} = 11(-1) = -11$.<br /><br />12. Simplify $i^{132}$<br /> Recognize powers of $i$: $i^{132} = (i^4)^{33} = 1$.<br /><br />13. Simplify $(7-3i)^2$<br /> Apply the formula $(a-b)^2 = a^2 - 2ab + b^2$: $(7-3i)^2 = 7^2 - 2(7)(3i) + (3i)^2 = 49 - 42i - 9(-1) = 58 - 42i$.<br /><br />14. Simplify $(4-3i)(7+i)$<br /> Use distributive property: $4(7+i) - 3i(7+i) = 28 + 4i - 21i - 3i^2$. Since $i^2 = -1$, it becomes $28 - 17i + 3 = 31 - 17i$.<br /><br />15. Simplify $(3i)^2$<br /> Calculate directly: $(3i)^2 = 9i^2 = 9(-1) = -9$.